404] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 445 



one range equals that between their correspondents in the other. But 

 this homography is only possible when a critical condition is fulfilled. 



In the first place, an infinite object on one range must have, as its 

 correspondent, an infinite object on the other. For if X be an infinite 

 object on one range, it has an infinite intervene with every other object 

 on that range (Axiom iv.) ; therefore X , the correspondent of X, must 

 have an infinite intervene with every other object on the second range. 



If, then, X and Y are the infinite objects on one range, and X and Y 

 the infinite objects on the other, and if A and B be two arbitrary objects on 

 the first range, and A and B their correspondents on the other; then, 

 using the accustomed notation for anharmonic ratio, 



But, if H be the factor (p. 440) for the second range, which H is for the first, 

 we have, since the intervenes are equal, 



H log (ABXY) = H log (A B X Y ) ; 

 and, since the anharmonic ratios are equal, we obtain 



H=H . 



If, then, it be possible to order two homographic systems of objects, so that 

 the intervene between any two is equal to that between their correspondents, 

 we must have H and H equal ; and conversely, when H and H are equal, 

 then equi-intervene homography is possible. 



We have therefore assumed Axiom v. ( 399) which we have now seen 

 to be equivalent to the assumption that the metric constant H is to be the 

 same for every range of the content. 



Nor is there anything in Axiom V. which constitutes it a merely gratuitous 

 or fantastic assumption. Its propriety will be admitted when we reduce our 

 generalized conceptions to Euclidian space. It is an obvious notion that 

 any two straight lines in space can have their several points so correlated 

 that the distance between a pair on one line is the same as that between 

 their correspondents on the other. In fact, this merely amounts to the 

 statement that a straight line marked in any way can be conveyed, marks 

 and all, into a different situation, or that a foot-rule will not change the 

 length of its inches because it is carried about in its owner s pocket. 



In a similar, but more general manner, we desire to have it possible, on 

 any two ranges, to mark out systems of corresponding objects, such that the 

 intervene between each pair of objects shall be equal to that between their 

 correspondents. We have shown in this Article that such an arrangement 

 is possible, when, and only when, the property v. is postulated. We may 

 speak of such a pair of ranges as equally graduated. 



